Javascript - easy, fast to use, large set of helpers and tools, no compile needed, runs natively in any modern browser, has the benefit of integrating HTML (I use it for UI elements) and allows you to insert code and fixes on the fly (fast and easy debugging) - Three.js is very popular, alternatively you can use just plain Javascript and WebGl, a lot of the syntax of which carry over into other languages.
Java - I'd argue obsolete, after all javascript now near meets it in power with the advent of HTML 5 and provides better support, but java is still a useful tool, its got a larger tool kit and an easier starting point, magnificent networking capabilities and one of the more enjoyable programming experiences - java is java, I advise you use java with java to java while you java, but I do believe it has some form of openGl available to it, for the sake of more graphically intensive programs (see minecraft)
C# - a powerful easy to learn language with a MAHOOSIVE tool set (the .net framework), it follows a similar syntax to many other programming languages, most notably the C and C++ it was based on - XNA is a good place to start, its where I started learning bigger things outside of game maker, alternatively there are very many engines, frameworks, independent libraries that can help you put together whatever kind of game you wish.
C++ - the industry standard, I won't sugar coat it, C++ is medium to learn very very hard to master, but it comes with outstanding perks, such vast expandability and used by more engines and frameworks than anyone would be insane enough to count, its performance can be next to none, you have the power, all of it, no hand holding, but likewise, no holding back - if you decide to go with C++ I'd advise you start from scratch, typically if you go for C++ your going for the real deal and you should probably get all the knowledge you can out of that, but if you want to start with an engine, Unreal 4 is now free open source and does some lovely tricks with C++

It's a high-level language which means that a lot of the dirty low-level stuff is taken care of. The OpenFl framework gives you the actual visual stuff, and allows you to do it in the way that Flash does - personally I find this a really nice way, although I started out with Flash so I suppose that's natural. Game engines like HaxePunk and HaxeFlixel are also good and do a lot of game-related things for you (like handling states, simplifying animation, etc). Despite all of this help it's still pretty flexible.
One big thing is also that it can compile to a range of platforms, natively. For desktop platforms you can only compile for the OS you're running (Windows to Windows, Linux to Linux, etc), but you can also compile to Android and iOS, as well as HTML5 and Flash. There's also Neko, which is kind of weird and it's not really obvious what it is, it's basically a target that runs in a virtual machine but compiles quickly and is similar to native desktop applications (usually you can also compile it to an .exe that runs without the virtual machine, although it does need some DLLs). Technically what Haxe it does is convert the code to whatever language (Objective C for iOS, Java for Android, etc) and then compile that, so you do need to download the SDKs or compilers for whatever you compile to (although it also helps you set this up).

There are a number of downsides. It is open-source and completely community-based, which means that things aren't really as stable and funded as, say, .NET languages. This also damages documentation; it can often be a bit hard to find any concrete answers to something that might not be clear, refined Googling skills are sometimes required. It's also relatively new, compared to stuff like C++ or Java, so again this affects stability and reliability. It's also not completely user-friendly to set up. I mean, the base language does have an installer, but the library handler, haxelib, is fully command-line, and any setup of libraries (like OpenFl and Flixel) are also done through the command-line. The actual compilation and stuff is also technically done through a console, but luckily FlashDevelop is an (unfortunately Windows-only) IDE that's pretty great and does help automate compilation. It's also just great for editing.

So yeah. If you wanna do some uncommon projects or whatnot things can get kinda messy (like trying to import external SWFs or make a 100% standalone Neko .exe) but if you just set everything up and start making games with something like HaxePunk or HaxeFlixel, you can usually ignore any nitty-gritty stuff. The language itself is, in my opinion, very nice to work with, it was developed by programmers for programmers and I can see that it was designed in that way. Plus, games seem to be the most supported type of project, so the road to making them is relatively smooth.